ecology letters, (2010) 13: 1182–1197 doi: 10.1111/j.1461 ... · mcmcglmm, population dynamics,...

R E V I E W A N DS Y N T H E S I S Causes and consequences of variation in plant

population growth rate: a synthesis of matrix

population models in a phylogenetic context

Yvonne M. Buckley,1,2� Satu

Ramula,3*� Simon P. Blomberg,1

Jean H. Burns,4 Elizabeth E.

Crone,5 Johan Ehrlen,6 Tiffany M.

Knight,7 Jean-Baptiste

Pichancourt,8 Helen Quested6

and Glenda M. Wardle9

Abstract

Explaining variation in population growth rates is fundamental to predicting population

dynamics and population responses to environmental change. In this study, we used

matrix population models, which link birth, growth and survival to population growth

rate, to examine how and why population growth rates vary within and among 50

terrestrial plant species. Population growth rates were more similar within species than

among species; with phylogeny having a minimal influence on among-species variation.

Most population growth rates decreased over the observation period and were negatively

autocorrelated between years; that is, higher than average population growth rates tended

to be followed by lower than average population growth rates. Population growth rates

varied more through time than space; this temporal variation was due mostly to variation

in post-seedling survival and for a subset of species was partly explained by response to

environmental factors, such as fire and herbivory. Stochastic population growth rates

departed from mean matrix population growth rate for temporally autocorrelated

environments. Our findings indicate that demographic data and models of closely related

plant species cannot necessarily be used to make recommendations for conservation or

control, and that post-seedling survival and the sequence of environmental conditions

are critical for determining plant population growth rate.

Keywords

Comparative analysis, demography, fire, herbivory, matrix population models,

MCMCglmm, population dynamics, population growth rate, spatial and temporal

variation, temporal autocorrelation.

Ecology Letters (2010) 13: 1182–1197

I N T R O D U C T I O N

In nearly all natural systems, environmental conditions

fluctuate over time. This environmental variation influences

population growth rates and consequently species abun-

dance and distribution. Environmental variation is typically

thought to elevate extinction risk. However, in perennial

plants with high adult survivorship, environmental variation

1School of Biological Sciences, University of Queensland,

Queensland 4072, Australia2CSIRO Sustainable Ecosystems, 306 Carmody Rd, St Lucia,

Queensland 4067, Australia3Section of Ecology, Department of Biology, University of

Turku, 20014 Turku, Finland4Center for Population Biology, University of California, Davis,

CA 95616, USA5Department of Ecosystem and Conservation Sciences, College

of Forestry and Conservation, University of Montana, Missoula,

MT 59812, USA

6Department of Botany, Stockholm University, Stockholm

10691, Sweden7Biology Department, Washington University in St. Louis,

St. Louis, MO 63130, USA8CSIRO Entomology, Indooroopilly, Qld 4068, Australia9School of Biological Sciences, University of Sydney, Sydney,

NSW 2006, Australia

*Correspondence: E-mail: [email protected]�These authors contributed equally to this work.

Ecology Letters, (2010) 13: 1182–1197 doi: 10.1111/j.1461-0248.2010.01506.x

� 2010 Blackwell Publishing Ltd/CNRS

that leads to rare episodes of high recruitment may promote

persistence (Higgins et al. 2000). Understanding how pop-

ulation growth rate varies in space and time in natural plant

populations, and identifying the sources of that variation is

fundamental to predicting species responses to environ-

mental change (e.g. climate change) and other environmen-

tal factors (Gotelli & Ellison 2006; Morris et al. 2008;

Dahlgren & Ehrlen 2009; Dalgleish et al. 2010).

Variation in population growth rates is due to differences

in underlying vital rates, such as birth, growth, reproduction

and death. Vital rates are in turn influenced by multiple

environmental factors (e.g. fire, herbivores, weather) and

their contributions to population dynamics depend on the

life history of the focal species (Silvertown et al. 1993, 1996;

de Kroon et al. 2000; Ramula et al. 2008a). Population

dynamics may vary substantially among populations and

years within the same plant species (e.g. Pascarella &

Horvitz 1998; Warton & Wardle 2003; Jongejans &

de Kroon 2005) and this variation may be related to

changes in population density. In general, we would expect

average population growth rates to be stable but patterns of

variation in population growth rates to differ among species.

Variation in population growth rates is likely to depend on

species life-form and lifespan (Garcıa et al. 2008; Dalgleish

et al. 2010), with herbs and grasses more likely to exhibit

variable population growth rates than trees and shrubs. If

similar environmental factors cause both spatial and

temporal variation, then understanding spatial variation in

population dynamics might allow understanding of temporal

variation, informing also under what circumstances (if any)

spatial dynamics might be substitutable for temporal

dynamics.

In randomly fluctuating environments, vital rates are

expected to vary from year to year in an unpredictable way.

However, in many natural environments fluctuations occur

non-randomly over a longer period of time causing

significant temporal autocorrelation. For example, succes-

sion results in a sequence of environmental conditions and

positive temporal autocorrelation in vital rates (e.g. Pasca-

rella & Horvitz 1998). Temporal autocorrelation can also be

negative such that a good year is followed by a bad year, for

instance, because of synchronized flowering and associated

costs of reproduction (Crone et al. 2009). Due to the rarity

of long-term demographic studies for plants (Menges 2000),

temporal autocorrelation is rarely included in predictions of

population performance (but see Pascarella & Horvitz 1998;

Quintana-Ascencio et al. 2003; Menges et al. 2006), although

in some cases, it has been shown to have a large effect on

predicted population dynamics (Pike et al. 2004; Tuljapurkar

& Haridas 2006).

Vital rates are most commonly linked to population

growth rate using matrix population models that describe

individuals classified by age, size or life-stage moving

through the life-cycle (Caswell 2001). Due to similar

construction and standard parameter estimation, matrix

population models have been widely used to quantify

population dynamics across plant species with different life-

histories, including rare and invasive species (e.g. Silvertown

et al. 1993, 1996; Ramula et al. 2008a; Burns et al. 2010). A

large and growing number of techniques exist for incorpo-

rating variation in vital rates into population growth rate

estimates (Tuljapurkar 1990; Fieberg & Ellner 2001; Kaye &

Pyke 2003; Tuljapurkar et al. 2003; Doak et al. 2005b;

Dahlgren & Ehrlen 2009). However, fewer studies have to

date examined the sources of variation and the importance

of realized levels of vital rate variation for population

growth rates of natural populations.

Closely related species often show more similarity in their

traits or geographic distributions than less related species

(e.g. Darwin 1859; Garland et al. 1993; Freckleton et al. 2002;

Blomberg et al. 2003; Cadotte et al. 2009; Diez et al. 2009),

and may therefore be expected to have more similar

population dynamics than non-related species, making

phylogeny a potential predictor of patterns of variation in

vital rates and population growth rates. If it is possible to

predict patterns of variation in population growth rate based

on species relatedness, phylogeny might be a useful tool for

identifying species that are likely to become endangered or

invasive, and for designing management when detailed

demographic information is not available. However, most

studies examining variation in plant population growth rate

have focused on a single or a few species and we do not

know enough about variation above the species level to be

able to say whether phylogeny is an important predictor of

population dynamics.

We constructed a database of demographic models with

both spatial and temporal replication from 50 species of

terrestrial plants and a corresponding phylogeny to examine

how potential sources of variation (e.g. vital rates,

phylogeny, life-form, rare or common distribution, envi-

ronmental factors) influence population growth rates. We

asked the following specific questions: (i) How do plant

population growth rates vary within species and among

species, and how does phylogeny contribute to that

variation? (ii) What are the sources of variation in

population growth rates? We examine how temporal

variation in vital rates (survival, growth, fecundity) contrib-

utes to population growth rate, and how rare or common

distribution, population density and environmental factors

(fire, herbivory) are related to variation in population

growth rate. (iii) What is the importance of realized

temporal variation in population growth rates to predicted

population dynamics? We compare deterministic and

stochastic population growth rates and examine the role

of temporal autocorrelation. If realized temporal variation

in population growth rates is unimportant, we would expect

Review and Synthesis Plant population dynamics in space and time 1183


average deterministic growth rate to be similar to stochastic

population growth rate.

We found that populations within a species are more

similar to each other than populations among species, with a

minimal influence of phylogeny above the species level. Our

findings indicate that the population growth rate of any

particular species cannot be substituted for that of a closely

related species to predict population dynamics. Most plant

populations studied had a decreasing trend in population

growth rates over the observation period and population

growth rates between years were negatively autocorrelated.

Population growth rates varied more through time than

space regardless of life-form. Temporal variation in a subset

of species for which we had environmental data was partly

explained by time-since-fire and herbivory. Deterministic

and stochastic population growth rates were highly corre-

lated for randomly fluctuating environments but differed for

temporally autocorrelated environments, particularly for rare

species with strong positive or negative autocorrelations.

Our results raise new questions about the causes and

consequences of temporal variation in vital rates, and

suggest that simple population models with uncorrelated

environmental variation may not adequately capture sto-

chastic population dynamics for strongly autocorrelated

environments.

M E T H O D S

We first summarize our database on matrix population

models and variables calculated to analyse population

dynamics. We then describe statistical analyses used to

answer the three main questions addressed above.

Database

The database, obtained from the literature and some

unpublished material from the authors and their collabora-

tors, contains matrix population models for 50 perennial

plant species in which multiple populations (‡ 2) and

multiple matrices per population (‡ 2) were available.

A total of 708 matrices for 38 herbaceous species (herbs

and grasses) and 120 matrices for 12 other species (trees,

shrubs and succulents) was assembled. This database is

unique as it includes both spatial and temporal population

dynamics for each species, allowing a detailed comparison

within and among species. Different subsets of the data

were used for analyses depending on which data were

available (detailed under specific analyses outlined below).

Species were categorized as common (including three

invasive species) or rare, or as common, restricted range

rare and wide range rare (i.e. locally rare but regionally

widespread), depending on the analysis. Species were

classified based on the author�s description. Matrix dimen-

sion was recorded at the species level as a factor with three

levels, small (£ 4 matrix dimensions), medium (£ 7) and

large (> 7); category boundaries were chosen to ensure

adequate replication within each category. Treating matrix

dimension as a categorical variable enabled simpler inter-

pretation of interactions than treating it as a continuous

variable with polynomial terms. Species were also classified

as clonal or not and having a seedbank or not. Spatial

locations of populations were sourced from the literature

where available and through personal communications with

the authors.

Calculation of variables

Population matrices were analysed using standard methods

(Cochran & Ellner 1992; Caswell 2001), implemented within

a custom built MATLAB program 7.1 (The MathWorks, Inc.,

Natick, MA, USA). In cases where the original papers

incorporated a seedbank incorrectly leading to a spurious

one-year delay in the life-cycle (Silvertown et al. 1993:

p. 467; Caswell 2001: p. 61), the matrix was corrected before

analysis was carried out. In a few species, some matrices

were reducible (Caswell 2001) because a matrix element

involving growth from one of the smallest stages was zero.

In these cases, we added a small value (10)20) to the matrix

(Ehrlen & Lehtila 2002).

Variation and sources of variation in population growth rates

For analyses of variation in population growth rates within

and among species, we used a subset of data for each

species in which the same consecutive years were surveyed

for every population (i.e. if some populations were studied

longer than others for a species, only a subset of the data

were used; this reduced our data set to 49 species, 197

populations, totalling 736 matrices; Table S1). This avoids

confounding spatial and temporal variation, gives a clearer

comparison of whether spatial or temporal variation in

population growth rates is greater and allows straightfor-

ward interpretation of temporal autocorrelation. We

calculated the long-term deterministic population growth

rate (log kdet) for each matrix according to Caswell (2001,

p. 108) and estimated standard deviation of population

growth rates across years for each population (rlog kdet).

Based on all annual matrices from each population, we

calculated standard deviation for five demographic rates

(hereafter denoted vital rates) across years: post-seedling

survival, growth to stages with a greater reproductive value

conditional on survival, retrogression to stages with a

lower reproductive value conditional on survival, the

number of seeds entering the seedbank and the number

of seedlings produced. Post-seedling survival was estimated

as the weighted mean of the survival of individuals not

included in the seed bank or the first non-seed stage (often

1184 Y. M. Buckley et al. Review and Synthesis


equivalent to the seedling stage). To take population

structure into account, we used the proportion of

individuals in the respective stage in the stable stage

distribution as weighting factors. For Taxus floridana, where

seedlings of different ages were present in multiple stages

(Kwit et al. 2004), we excluded only the first of these

stages in a matrix for our estimates of post-seedling

survival. Similar to post-seedling survival, we excluded the

first non-seed and non-dormant stage (often the seedling

stage) in each matrix from the calculations of growth and

retrogression. We used average variation of growth and

retrogression (hereafter denoted growth) to avoid losing

matrices and species with missing growth or retrogression

transitions. Since actual seed and seedling production

varied greatly among species, we calculated the coefficient

of variation for both these variables to standardize the

scale (Sokal & Rohlf 1995; CV for small samples eq. 4.9).

We used average variation in seed and seedling production

when both variables occurred simultaneously in a matrix;

otherwise variation in either seed or seedling production

was used depending on which variable was presented for a

given species.

As the exact location of populations was rarely available,

we used the approximate distance between populations for

each species (available for 43 species; Table S1) to examine

whether populations in close proximity were more similar

in population growth rate than more distantly located

populations. Where available we calculated average plant

density per m2 for each population and year (21 species)

and noted habitat type and environmental factors (time-

since-fire and herbivory). Data on environmental factors

affecting population growth rate were only available for a

few species, time-since-fire was recorded for three species

and herbivory intensity was recorded for four species

(Table S4).

Importance of temporal variation in plant population growth rates

The full 50 species dataset and all available populations with

multiple years were used. For each population of each

species, we constructed a mean matrix by averaging multiple

annual matrices and calculated deterministic population

growth rate (log kmean). This estimate is based on average

vital rates over time and can sometimes be used to describe

average population performance (Doak et al. 2005a).

However, mean matrix population growth rate often differs

from the long-term stochastic population growth rate

(Tuljapurkar et al. 2003; Boyce et al. 2006), and differences

between these two rates can be regarded as a measure of the

influence of temporal variation in vital rates on population

growth rates.

To examine the importance of temporal variation in plant

population growth rates to predicted population dynamics,

we compared log kmean to stochastic population growth

rates calculated for randomly fluctuating environments. We

calculated stochastic population growth rates for each

population using a simulation which started from the stable

stage distribution derived from the mean matrix with 1000

individuals. All simulations were based on a matrix selection

method which is suitable for a small number of matrices

(Ramula & Lehtila 2005). In these simulations, each matrix

had an equal probability of being selected at each time

step and population dynamics were predicted for 500 years

with 10 000 iterations per year. Stochastic population

growth rate (ks) was calculated as log[(Nt ⁄ N0)1 ⁄ t ], where

N is population size at time t (Caswell 2001). In addition to

the simulations, we used Tuljapurkar�s analytical approxi-

mation for uncorrelated environments (Caswell 2001;

eq. 14.72), which estimates the stochastic population growth

rate (kTulja) from observed matrices and covariances among

matrix entries. This method provides an alternative to

simulations but may not describe population dynamics

accurately if there is a lot of variation in vital rates over time

(Morris & Doak 2002).

Both the simulation and Tuljapurkar�s method ignore

temporal autocorrelation, assuming uncorrelated population

growth rates over time. To estimate population growth rates

for temporally autocorrelated environments, we used matri-

ces from each population in the same sequence they were

observed in the field and calculated population growth rate

(kseq) by taking the n-th root of the greatest positive

eigenvalue of the matrix product [AnAn)1…A1] where A

denotes annual matrices over time and n the total number of

matrices per population. Throughout the paper, we use log-

transformed mean matrix growth rates to enable compari-

sons between them and stochastic population growth rates,

with population growth rates < 0 denoting decreasing

populations and population growth rates > 0 denoting

increasing populations.

Analyses

Variation in population growth rates within species and among species

To examine the effect of phylogeny on variation in

population growth rates, we generated a phylogeny using

the seed plant phylogeny available via phylomatic (reference

tree #R20050610), which references the angiosperm phy-

logeny website, the most recent, and constantly updated,

summary of the angiosperm phylogeny available (phylomatic

version 2; Webb C.O. 2005; Webb et al. 2008; Stevens 2009).

This topology was then assigned branch lengths based on

fossil calibration following Wikstrom et al. (2001) using the

bladj command in phylocom (version 4.0.1b), which approx-

imates branch lengths in millions of years (Webb et al. 2008;

see Fig. S1).

As we had multiple levels of random effects: species

within a phylogeny, populations within species and years



within populations, we used the R library MCMCglmm

v.1.10 to test for effects of phylogeny. MCMCglmm is a

Markov chain Monte Carlo sampler for multivariate

generalized linear mixed models with special emphasis on

correlated random effects arising from pedigrees and

phylogenies (Hadfield & Nakagawa 2010). MCMCglmm

enables the inclusion of a phylogeny, equivalent to a

pedigree in quantitative genetics studies, as a variance ⁄co-variance matrix in a generalized linear modelling Bayes-

ian framework. The MCMCglmm model also enables

random effects to be fit at the species and below species

levels. We fit species and populations nested within species

as random effects for models of deterministic population

growth rate (log kdet), and a random effect of species for

models of standard deviation of population growth rate

(rlog kdet) and comparisons of mean matrix and stochastic

population growth rate. We used weak proper priors on the

grouping variables (variance v > 0, degree of belief n = 1)

and assessed an arbitrary range of different values for the

prior variances (0.001 ) 1). We also assessed the effect of

prior selection on datasets where the response variable was

randomized. The adequacy of models with and without

phylogeny was assessed by comparing the Deviance

Information Criterion (DIC) of the two models (Spiegel-

halter et al. 2002).

The phylogenetic effects have expected (co)variances

proportional to the phylogenetic covariance matrix con-

structed from the phylogenetic tree (Fig. S1) for the

MCMCglmm models (Hadfield & Nakagawa 2010). The

within group errors were assumed to be independent and

identically normally distributed (IID), with mean 0 and

variance r2, except where a within group correlation

structure was explicitly applied. The random effects were

also assumed to be IID for different groups, with mean 0

and covariance matrix w. IID assumptions for residuals and

random effects were assessed using plots of the within

group errors and random effects (Pinheiro & Bates 2000:

p. 174–196). Chains mixed well with little autocorrelation,

indicating good convergence.

We examined the effect of phylogeny on population

growth rate (log kdet) using a model which included all

predictors (see Table 1) as well as a simpler version of the

model which only retained significant terms from the non-

phylogenetic models (see below). Autocorrelation of errors

to account for autocorrelation between time periods within

a population could not be fitted in the MCMCglmm

models for testing phylogeny and was therefore omitted.

Similarly, we examined the effect of phylogeny on standard

deviation of population growth rate (rlog kdet) and

stochastic population growth rates (ks, kTulja, kseq) using

a model which included several predictors and variance

at the species level (see Tables 2 and 3 for predictors

fitted).

Sources of variation in population growth rates

As the inclusion of phylogeny was not supported for

models of population growth rate (log kdet) (see Results and

Table S2), we used models without phylogeny to test

hypotheses about the structure and partitioning of variation

between spatial and temporal components for log kdet.

Simple variance components analysis of log kdet was used

initially to assess the effects of population level vs. year level

random effects (nested within species) to determine the

relative contributions of spatial vs. temporal effects.

Population and year effects could not be included in the

same model due to the lack of replicates at the within

population, within year level. As population level variation

may partly depend on spatial distances among the study

populations with populations closer together exhibiting

more similar dynamics, we also fitted models only including

species with populations at least 1 km apart (24 species) to

explore the effect of spatial distances among the popula-

tions on variation in population growth rates (i.e. whether

the results differ from those when all populations are

included).

To better model temporal effects on population growth

rate, we constructed a general linear mixed effects model

which included a fixed effects structure, a random effects

structure and autocorrelation between errors for sequences

of years within a population. For all of the following

non-phylogenetic linear mixed effects models, random

Table 1 Predictor variables and their significance in non-phyloge-

netic models for deterministic population growth rate, log kdet.

Variables retained in the simplest adequate model are in bold.

Fixed effects were assessed using likelihood ratio tests and

maximum likelihood estimates. Random effects were assessed

using likelihood ratio tests and restricted maximum likelihood

estimates. The random effect of species was not tested as

populations were nested within species and were retained in the

model. Data were from 49 species

Predictor variables LR testd.f., P-value

Year LR1 = 10.7, < 0.005

Matrix dimension (small, medium, large) LR2 = 0.4, < 0.85

Life-form (herb, other) LR1 = 0.7, < 0.5

Clonality (binary) LR1 = 0.1, < 0.96

Seedbank (binary) LR1 = 1.5, < 0.3

Distribution (common, rare restricted,

rare widespread)

LR2 = 0.1, < 0.95

Year : life-form LR1 = 2.1, < 0.2

Year : clonality LR1 = 3.2, < 0.3,

Year : seedbank LR1 = 2.5, < 0.2

Year : distribution LR2 = 2.6, < 0.3

r2species (intercept) Not tested

r2population (intercept) LR2 = 39.5, < 0.0001

r2population (slope of year) LR2 = 11.4, < 0.004

Autocorrelation of residuals (AR1) LR1 = 20.7, < 0.0001



effects were estimated using restricted maximum likelihood

(REML) and likelihood ratio (LR) tests, where the v2

distribution was used to determine P-values. Where the

errors from sequences of years within a population were

assumed to be distributed according to an auto-regressive

process of order 1 (AR1) (i.e. for temporal autocorrelation),

autocorrelation of errors was tested using REML tests

between models with and without the autocorrelation. Fixed

effects were assessed using maximum likelihood and

likelihood ratio tests (Bolker et al. 2009). We used the nlme

library in R (v. 2.9.2) (Pinheiro et al. 2009) to include

autocorrelation between errors for consecutive years.

Table 1 shows the variables fitted for population growth

rates (log kdet). We tested the slope of year as a fixed effect

and variance about the slope of year as a random effect. The

random effects structure included species, populations

nested within species and variance about the slope of year

at the population level with temporal autocorrelation

between errors (AR1). To examine the consistency of

autocorrelation for longer time series, we tested autocorre-

lation between errors (AR1) using a subset of our data that

contained populations with ‡ 3 consecutive year matrices

per population (610 matrices from 28 species). This subset

of data produced qualitatively and quantitatively similar

results to the full data set and we therefore report results

based on the full dataset only.

We examined the roles of vital rates, environmental

factors, habitat and population density as drivers of

population growth rates (log kdet). To investigate direct

relationships between population growth rate (log kdet) and

the vital rates of survival, growth and fecundity, we used a

model with autocorrelation of errors (31 species, 601

matrices due to missing data). We tested fire and herbivory

as drivers of variation in population growth rates using data

from three species (35 populations) for fire and four species

(25 populations) for herbivory. Time-since-fire or herbivory

(as linear and quadratic effects) was used as an explanatory

variable in linear mixed effects models to test for its effect

Table 2 Predictor variables and their significance in non-phyloge-

netic models for standard deviation of population growth rate,

rlog kdet. Variables retained in the simplest adequate model are in

bold. Fixed effects were assessed using likelihood ratio tests and

maximum likelihood estimates. Random effects were assessed

using likelihood ratio tests and restricted maximum likelihood

estimates. The random effect of species was not tested as

populations were nested within species and were retained in the

model. There were missing data for Campanula americana, Alnus

incana and Astragalus alopecurus, models were therefore run including

47 species. Note that simplified models including log rgrowth

instead of log rsurvival were structurally identical; however, models

with log rsurvival were strongly preferred to models with

log rgrowth (DAIC > 10)

Predictor variables LR testd.f., P-value

Log kmean LR1 = 10.1, < 0.002

Matrix dimension (small, medium, large) LR1 = 4.3, < 0.2

Distribution (common, rare restricted,

rare widespread)

LR1 = 0.01, < 0.95

No. matrices per population LR1 = 6.0, < 0.015

Life-form (herb, other) LR1 = 0.2, < 0.7

Seedbank (binary) LR1 = 0.8, < 0.4

Log rsurvival LR1 = 28.0, < 0.0001

CV fecundity LR1 = 1.7, < 0.2

r2species (intercept) Not tested

r2population (intercept) Not tested

r2population (slope of log rsurvival) LR2 = 13.6, < 0.0015

Table 3 Non-phylogenetic models for three measures of stochastic population growth rate (ks, kTulja and kseq) for 50 species. Variables

retained in the simplest adequate model are in bold. Fixed effects were assessed using likelihood ratio tests and maximum likelihood estimates.

Random effects were assessed using likelihood ratio tests and restricted maximum likelihood estimates. Some explanatory variables were not

tested because of significant interactions

Predictor variables ks kTulja kseq

Log kmean Not tested LR1 = 111.2, P < 0.001 Not tested

Matrix dimension (small, medium, large) LR2 = 8.2, P < 0.02 LR2 = 6.3, P < 0.05 LR2 = 6.4, P < 0.05

No. years per population LR1 = 1.35, P < 0.3 LR1 = 0.9, P < 0.4 LR1 = 1.8, P < 0.2

Life-form (herb, other) LR1 = 0.26, P < 0.7 LR1 = 0.26, P < 0.7 LR1 = 0.3, P < 0.6

Seedbank (binary) LR1 < 0.0001, P < 1 LR1 = 0.07, P < 0.8 LR1 = 3.4, P < 0.07

Distribution (common, rare) Not tested LR1 = 8.2, P < 0.005 Not tested

Log kmean : no. years per pop LR1 = 0.04, P < 0.85 LR1 = 1.3, P < 0.3 LR1 = 1.7, P < 0.2

Log kmean : matrix dimension LR2 = 0.75, P < 0.7 LR2 = 4.0, P < 0.2 LR2 = 0.08, P < 0.96

Log kmean : life-form LR1 = 0.4, P < 0.6 LR1 = 0.64, P < 0.5 LR1 = 0.04, P < 0.9

Log kmean : seed bank LR1 = 0.21, P < 0.7 LR1 = 1.8, P < 0.2 LR1 = 1.3, P < 0.3

Log kmean : distribution LR1 = 5.0, P < 0.03 LR1 = 2.3, P < 0.2 LR1 = 13.1, P < 0.001

r2species (intercept) Not tested Not tested Not tested

r2species (slope of log kmean) LR2 = 20.7, P < 0.001 LR2 = 33.8, P < 0.001 LR2 = 56.7, P < 0.001



on population growth rate across species. We also tested

for autocorrelation between years. As time-since-fire and

year were confounded, we did not include year in the fire

model. For the herbivory model, we included year as a fixed

effect and variance about the slope of year as a random

effect to account for possible temporal trends. We note that

all three species with data available for fire were from the

same geographic region (Central Florida, USA). We

examined the effect of habitat on log kdet. Habitat data

were available for 483 matrices and two categories of

habitat had sufficient species level replicates, grassland

(15 species, 53 populations) and forest (18 species, 86

populations). Finally, we tested whether density affected

log kdet for a subset of 207 matrices (21 species, 58

populations). We constructed models including density, year

and random effects of species and population (with

variance about the intercept and slope of density at the

population level to account for context specific effects of

density) with autocorrelation of errors (AR1) for sequences

of years within populations.

For analyses of standard deviation in population growth

rate, rlog kdet, we used the same predictors as for analyses of

log kdet (see Table 2 for predictors). In addition, we

examined the contributions of temporal variation in vital

rates (standard deviation of survival and growth and CV of

fecundity) in a subset of species (n = 47) for which these

data were available. Plots and initial modelling indicated

that variation in survival and growth (log rsurvival and

log rgrowth) were co-linear, therefore the sources of variation

in population growth rates was analysed using a model which

included the predictors in Table 1 and either log rsurvival or

log rgrowth; the better predictor was assessed using AIC. The

random effects structure included variance about the

intercept and about the slope of log rsurvival or log rgrowth

at the species level. As phylogeny was not important, we used

linear mixed effects models (nlme library in R).

Importance of temporal variation in population growth rates

To examine the importance of realized temporal variation in

plant population growth rate for randomly fluctuating

environments, we compared mean matrix population

growth rate (log kmean) to stochastic population growth

rates (ks and kTulja), and further examined the role of

temporal autocorrelation in plant population dynamics by

comparing mean matrix population growth rate to kseq. As

the inclusion of phylogeny was not supported in any of the

models (see Results), we used linear mixed effects models

(lme4 library in R to ensure model convergence) with

variance about the intercept and the slope of mean matrix

population growth rate (log kmean) with species as a random

effect to estimate model parameters (Table 3).

To determine whether the magnitude of the temporal

autocorrelation explained differences between stochastic

population growth rates for randomly fluctuating and

temporally autocorrelated environments, we constructed a

model for a subset of species with at least three consecutive

matrices (n = 26) with kseq as a response variable and the

following explanatory variables: ks, temporal autocorrelation

of annual population growth rates (as linear and quadratic

effects), distribution (rare, common) and their interactions.

Species was used as a random effect.

Normality and homogeneity assumptions were examined

for each model visually from residual plots and were well

supported. Exceptions were models of stochastic popula-

tion growth rates, where the differences between mean

matrix and stochastic population growth rates were mostly

in one direction only, leading to asymmetric distribution of

residuals. However, fitting heterogeneous variances by

species distribution (common ⁄ rare) or matrix dimension

was not supported by LR tests, indicating that the model

assumptions about homogenous variances were not severely

violated.

R E S U L T S

Variation in population growth rates within speciesand among species

The effect of phylogeny on all response variables was weak,

i.e. the phylogenetic relationship between species was

unrelated to deterministic population growth rate, standard

deviation of population growth rate, or the relationship

between mean matrix and stochastic population growth

rates. In analyses of deterministic population growth rate

(log kdet), the non-phylogenetic model was preferred

(DDIC > 7.5 for all priors tested). The proportion of

variance explained by phylogeny was very sensitive to the

priors selected, ranging from < 1 to 28% for the model of

log kdet and was similar to that for randomized log kdet

where phylogeny should convey no information (< 1–18%;

Table S2). A value of 0 indicates the grouping conveys

no information, up to 100% where all members of a group

are identical (Gelman & Hill 2007). The proportion of

variance explained by species was insensitive to prior

selection (c. 14% for all priors tested). Similarly, in analyses

of the standard deviation of population growth rate and

stochastic population growth rates, the models without

phylogeny were either identical to the models with

phylogeny (DDIC < 1) or models without phylogeny were

strongly preferred (DDIC > 6.5).

Sources of variation in population growth rates

Populations within a year were more similar to each other

than the same population across years, although this pattern

depended on spatial scale and temporal autocorrelation.



Variance components analysis of population growth rate

used to determine spatial and temporal effects revealed that

the population level effect when nested within species was

small (missing middle grey bar in Fig. 1a), whereas when

year was nested within species it explained a similar

amount of variance in population growth rates as species

(Fig. 1a). However, when species with populations ‡ 1 km

apart were analysed, the proportion of variance explained

by population increased (middle grey bar in Fig. 1b),

indicating that population growth rates were more similar

within a population through time, i.e. the year effect

weakened and the population effect strengthened. For both

cases, the inclusion of autocorrelation between years

increased the explanatory power at the population

level and reduced the residual variance (black bars in

Fig. 1a,b).

The effect of year on population growth rate (log kdet)

was significant and negative, indicating a decreasing trend in

population growth rate over time (Table 1, Fig. 2 for

predicted values, Table S3 for parameter estimates). Popu-

lation growth rates decreased through time at different rates

for different populations (Fig. 2); variance about the slope

for year within population was significant (Table 1). Despite

the overall negative trend in population growth rate through

time most populations remained increasing (log kdet > 0)

throughout the studies (Fig. S2). Around these average

trends in population growth rates, autocorrelation of annual

residuals was highly significant (Table 1) and negative

(u = )0.3), meaning that years with higher than average

population growth rates tended to be followed by years with

lower than average population growth rates and vice versa.

When autocorrelation of errors within populations and

population-specific temporal trends were included in the

model, the proportion of variance explained at the

population level increased substantially relative to the simple

variance components analysis (Fig. 1a,b). Appropriate mod-

elling of within population temporal processes was therefore

important.

To explore possible mechanisms for temporal autocor-

relation in annual population growth rates, we examined

autocorrelations (AR1, using the acf function in R) for 133

populations with > 3 years of data from consecutive years

and found that autocorrelations in survival (LR1 = 14.2,

P < 0.0005) and fecundity (LR1 = 8.56, P < 0.004) had

significant additive positive contributions to autocorrela-

tions in population growth rate.

Examination of direct relationships between vital rates

and population growth rate revealed that log(survival)

contributed to population growth rate significantly when

tested in a model including autocorrelation of errors and

varying slopes for survival (LR1 = 68.6, P < 0.0001);

varying slopes for year could not be included due to model

convergence issues. Log(growth) was significant when

included in a model with autocorrelation and varying slopes

for year (LR1 = 5.97, P < 0.02). Fecundity was not

significant.

Both time-since-fire and herbivory intensity explained part

of the temporal variation in population growth rate. There

was a significant effect of the quadratic term of time-since-

fire (LR1 = 19.5, P < 0.0001), with no significant population

level variation in the slope (LR1 = 0.0003 P > 0.9) (Fig. 3).

This indicates that population growth rates can be expected

to decline after fire until reaching more stable dynamics.

There was also a significant negative autocorrelation between

years, similar in magnitude to the autocorrelation found in

the full dataset (u = )0.38, LR1 = 15.8, P < 0.0001).

Population growth rate declined linearly with increasing

herbivory intensity (herbivory: LR1 = 9.8, P < 0.002 and

year: LR1 = 6.59, P < 0.02, Fig. 4). Autocorrelation of

errors (LR1=0.6, P > 0.5) and population level random

effects (LR1=0.05, P > 0.9) were not statistically significant.

Models with and without habitat were very similar

(DAIC = 0.6), with little support for retaining habitat in

YearPopulationAutocorr

Species0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Pro

port

ion

of v

aria

nce

expl

aine

d

Year/population Residual

(a)

(b)

Figure 1 Variance components analysis of deterministic popula-

tion growth rate (log kdet) for three models: (i) years nested within

species (YEAR), (ii) populations nested within species (POPULA-

TION) and (iii) population nested within species, in a model

including autocorrelation of errors between sequential years

(AUTOCORR). (a) Data from 49 species and (b) data from 24

species with populations ‡ 1 km apart.



the model for this subset of the data. Finally, our analyses

revealed no evidence for density dependence in population

growth rates. The random effect of varying slopes for density

at the population level was not significant (LR2 < 0.001,

P > 0.9) and a fixed effect of density and habitat were also

not significant (LR1 = 0.8, P < 0.4 and LR1 = 2.6, P < 0.1,

respectively).

Variation in survival and growth rather than variation in

fecundity explained temporal variation in population

growth rates (rlog kdet). Variation in rlog kdet increased

with mean population growth rate (log kmean), the number

of matrices and the standard deviation of survival

(log rsurvival) (Fig. 5, Table 2, Table S3 for parameter

estimates).

Importance of variation in population growth rates

Mean matrix population growth rate (log kmean) was a good

predictor of all measures of stochastic population growth

rate (ks, kTulja and kseq) (Table 3). However, mean matrix

population growth rate tended to be greater relative to ks

and kTulja for rare species and species with medium and

small matrix dimensions (Fig. 6). Departure from a 1 : 1

relationship with mean population growth rate was most

1 3 5

–0.2

A. spicata

1 2 3 4 5

–0.1

0

A. eupatoria

1.0 2.0 3.0

–0.1

5

A. incana

3.0 4.5 6.0

0.0

A. fecunda

1.0 2.0 3.0

0.00

A. elliptica

1.0 1.4 1.8

0.00

A. triphyllum

1.0 2.5 4.0

–0.1

A. bipartita

1 2 3 4 5

–0.4

A. canadense

1.0 2.5 4.0

0.0

A. alopecurus

1 2 3 4 5

–0.2

A. scaphoides

2 4 6 8

–0.6

A. tyghensis

1.0 1.4 1.8–0.0

05

B. excelsa

1.0 2.5 4.0

–0.2

C. ovandensis

1.0 1.4 1.8

1.0

C. americana

1.0 2.0 3.0

–1.0

C. keyensis

1.0 2.5 4.0

–0.2

C. elata

1.0 2.0 3.0

–0.2

C. palustre

1.0 2.0 3.0

0.1

C. hirta

1.0 1.4 1.80.

0

C. scoparius

1.0 1.4 1.8

–0.4

D. sericea

1.0 1.4 1.8

–0.3

D. purpurea

1.0 1.4 1.8

–0.0

6

E. latifolium

1 3 5

–0.2

E. cuneifolium

1.0 1.4 1.8

0.00

E. edulis

1.0 1.4 1.8

0.00

G. reptans

1.0 2.0 3.0

–0.0

6

G. rivale

2 4 6 8

–0.5

H. radiatus

1.0 1.4 1.8

–0.0

1

H. acuminata

1 2 3 4 5

–0.5

H. cumulicola

1.0 2.0 3.0

–0.1

L. vernus

1 3 5 7

–1.0

L. bradshawii

1 2 3 4 5

–0.4

L. cookii

1.0 1.4 1.8

–0.0

5

M. magnimamma

1.0 2.0 3.0

–0.6

M. cardinalis

1.0 2.0 3.0

–0.4

M. lewisii

1.0 1.4 1.8

0.00

P. quinquefolium

1.0 2.5 4.0

–0.1

4

P. media

1.0 2.0 3.0

–0.1

5

P. farinosa

2.0 2.4 2.8

–0.3

P. veris

1.0 1.4 1.8

0.0

P. vulgaris

1.0 1.4 1.8

–0.0

4

P. gaumeri

3.0 4.0 5.0–0.0

40

P. subintegra

1.0 2.5 4.0

–0.1

5

S. europaea

1.0 1.4 1.8

–0.0

6

S. pulcherrima

1.0 2.5 4.0

–0.2

S. cotyledon

1.0 2.0 3.0

–0.3

S. pratensis

1.0 2.5 4.0

–0.0

35

T. floridana

1.0 1.4 1.8

–3.0

T. pratensis

2.0 3.0 4.0

–0.1

5

T. grandiflorum

Year

Log

(pop

ulat

ion

grow

th r

ate,

λde

t)

Figure 2 Deterministic population growth

rate (log kdet) for 49 species, points are

observed data from multiple populations

and lines are the predicted values for each

population from the multi-level mixed

effects model (Table 1).



noticeable for rare species where stochastic population

growth rate was calculated taking autocorrelation into

account (kseq) (Fig. 6). Mean matrix and stochastic popula-

tion growth rates sometimes predicted opposite population

dynamics; for 14 of 222 populations (6%) log kmean

indicated an increasing population, whereas kseq indicated

a declining population and vice versa for four of 222

populations (2%).

The contribution of temporal autocorrelation to kseq was

nonlinear (LR1 = 9.3, P < 0.01 for the ks · quadratic auto-

correlation interaction) and depended on species distribution

and the magnitude of ks (LR1 = 20.9, P < 0.0001 for the

distribution · ks · autocorrelation interaction). Both large

positive and negative autocorrelations caused departure from

an expected 1 : 1 relationship between ks and kseq especially

for rare species (Table S6 for parameter estimates), showing

that the strength of autocorrelation is important. In our

dataset, the six data-points that departed most strongly from

the expected 1 : 1 relationship had large negative autocorre-

lations and for the five rare species ks was substantially higher

than kseq (Fig. 7).

D I S C U S S I O N

Plant population growth rates exhibit a signal of species

For our dataset of 50 perennial plants, we found species to

be an important predictor of population dynamics with

phylogeny above the species level having a little or no

explanatory power. This indicates that patterns of variation

in plant population growth rates are evolutionarily labile

above the species level for the species sampled in this study,

suggesting that we cannot predict population dynamics

based on phylogenetic relatedness alone. Although our

database is taxonomically sparse with few species per genus,

other studies using larger databases have also reported a

negligible effect of phylogeny on the evolution of plant life-

histories, vital rates, and hydrological niches in plant

Chamaecrista keyensis

Time-since-fire

Eryngium cuneifolium

0 5 10 15 20 25

5 10 15 20 25

30

0 5 10 15 20 25 30

–0.5

–0.6

–0.2

–0.4

–0.2

0.0

0.2

0.4

0.0

0.2

0.4

0.0

0.5

1.0

1.5

Hypericum cumulicola

Log

(pop

ulat

ion

grow

th r

ate,

λde

t)

Figure 3 Effect of time-since-fire on population growth rate

(log kdet) for three species. Points are k estimates for each matrix

and the lines are fitted values with the BLUPs for each species.

Population level variation in the intercept was significant (LR1 = 5.4,

P < 0.03) but very low and for clarity is not shown here.

1 2 3 4 5 6

Actaea spicata

Year

MinMeanMax

Cirsium palustre

2.0 2.2 2.4 2.6 2.8 3.0 1.0 1.5 2.0 2.5 3.0

1.0 1.5 2.0 2.5 3.0

3.5 4.0

–0.0

5

–0.0

50.

05–0

.2–0.2

–0.1

0.1

0.0

0.2

0.0

0.4

–0.1

5

0.05

0.15

Lathyrus vernus Trillium grandiflorum

Log

(pop

ulat

ion

grow

th r

ate,

λde

t)Figure 4 Effect of year and herbivory on population growth rate

(log kdet) for four species. Lines are predicted values (BLUPs) for

min, mean and max herbivory (%) within each species (see

Table S5 for min, mean and max values).



communities (Silvertown et al. 2006; Kuster et al. 2008;

Burns et al. 2010); therefore, we consider the lack of a

phylogenetic signal on population dynamics found here to

be biologically reasonable. We note however that a more

fine scale sampling of genera (i.e. a larger number of species

per genus) could still reveal a phylogenetic signal on patterns

of variation in population growth rates.

Although several studies have shown that phylogenetic

relatedness or taxonomic groupings can be useful for

predicting rarity or invasiveness (Pysek 1998; Daehler 1998;

Daehler et al. 2004; Pheloung et al. 1999; Purvis et al. 2000;

Schwartz & Simberloff 2001; Diez et al. 2009; but see

Lambdon 2008), our results indicate that this does not likely

result from phylogenetic constraints on demography. Our

results suggest that demographic data of closely related

species may provide little information on demography for

the management of rare or invasive species.

Plant population growth rates decline through time

We found a negative temporal trend in population growth

rates for most of the populations. This trend held regardless

–8 –6 –4 –2

–8–6

–4–2

0

2 3 4 5 6 7 8 9

–8–6

–4–2

0

# matrices

–0.5 0.0 0.5

–8–6

–4–2

0

Log λmean

Log σsurvival

σ lo

g λ d

etσ

log λ d

etσ

log λ d

et

(c)

(b)

(a)

Figure 5 Temporal variation in population growth rate is partly

explained by (a) standard deviation of survival (log rsurvival), (b) the

number of matrices per population and (c) mean matrix population

growth rate (log kmean). Points are observed data from populations

within 47 species.

CommonRare

–0.5

0.0

0.5

1.0

–0.5

0.0

0.5

1.0

–0.5

0.0

0.5

1.0

1.5

–0.5

0.0

0.5

1.0

1.5

–0.5

0.5

1.0

–0.5

0.5

1.0

SmallMediumLarge

–0.5 0.0 0.5 1.0 –0.5 0.0 0.5 1.0

–0.5 0.0 0.5 1.0 –0.5 0.0 0.5 1.0

–0.5 0.0 0.5 1.0 –0.5 0.0 0.5 1.0

λ seq

λ Tul

ijaλ s

Log λmean

(c)

(a)

(d)

(e) (f)

(b)

Figure 6 Relationships between mean matrix population growth

rates (log kmean) and stochastic population growth rates (ks, kTulja

and kseq) for 50 species. The solid lines are expected 1 : 1

relationships between the population growth rates if log kmean and

the stochastic ks were exactly equivalent. Each row represents a

different stochastic population growth rate (a & b ks, c & d kTulja,

e & f kseq), the first column uses different symbols for common

and rare species (a, c, e) and the second column uses different

symbols for matrix dimensions (small, medium & large) (b, d, f).



of common or rare status; 51% of populations had positive

population growth rates at the beginning of a study period

but this reduced to 42% by the end of the study period. We

offer four possible explanations for this overall negative

trend in population growth rates, two of which reflect

sampling artefacts and two of which reflect actual declines in

abundance. The most obvious artefactual explanation would

be if the sampling process itself caused deterioration of the

vital rates determining population growth rate. We think this

is unlikely for studies of plant demography because these

typically involve minimal handling (few days per year).

Second, demographers may tend to start studies in

�best� sites with relatively high population densities,

increasing the likelihood of population declines. For

example, Bierzychudek (1999) identified density-dependent

population dynamics as a potential source of an unpredicted

population decline. However, in our dataset increasing and

decreasing populations were both equally likely at the

beginning of a time period (100 populations log kmean > 0

and 97 populations log kmean < 0), which is what we would

expect from a random sample of populations with stable

dynamics (log kmean = 0).

Therefore, we suggest that this general pattern reflects

more subtle causal factors. Researchers may choose to

terminate studies after a particularly bad year or run of years

leading to estimated declines in population growth rate.

Finally, it may be that habitat quality itself tends to fluctuate

over time. Fluctuating environmental conditions could

reflect successional dynamics (Menges 1990) or changes in

microbial communities, e.g. accumulation of pathogens or

less beneficial mychorrizae (Bever 1994). If demographers

tend to start studies in �best� sites with relatively high

population densities, then trends over relatively short

time periods would be biased toward declines. If this

mechanism were true, it would be necessary to establish

study plots or transects not only in the central parts of the

populations but also at the edges (i.e. sparser locations)

that might be those �best� locations in the future, to detect

true population-level trends. Further study of the charac-

teristics and locations of the studied populations may be

able to tease out broad landscape scale patterns correlated

with declining population growth rates (e.g. particular

land-use types or habitats). If sampling bias causes an

overly pessimistic view of population persistence, efficient

and effective conservation and restoration actions may be

compromised as, in the worst case, unnecessary manage-

ment will be implemented.

Sources of variation in plant population growth rates

Quantifying variation in plant population dynamics across

spatial and temporal scales is important for predictions of

species abundance and distribution. Our comparative study

based on a large number of plant species across different

habitats revealed quantitative and qualitative differences

between spatial and temporal variation in plant population

dynamics. Population growth rates varied more between

years than between populations, although the analysis of

populations > 1 km apart reduced the difference between

spatial and temporal variation, showing that greater temporal

variation in plant populations is partly scale-dependent and

caused by closely situated study populations within species.

Earlier studies have detected synchronized population

dynamics for short-lived grassland species (populations

situated within a few kilometers), with the magnitude of

synchrony clearly decreasing with increasing spatial distance

among populations (e.g. Ramula et al. 2008b; Kiviniemi &

Lofgren 2009). Our results suggest that synchronized

population dynamics might occur also for longer-lived

perennials, for instance, due to similar environmental factors

in closely situated populations. Unfortunately, the level of

detail on spatial locations reported was not adequate for us

to construct a spatial variance–covariance matrix to further

explore how distance between populations affects population

growth rate and this remains a question for future studies.

Observed temporal variation in population growth rates

was mainly due to variation in post-seedling survival

regardless of the life-form; a typical pattern for long-lived

perennials (Silvertown et al. 1993, 1996; de Kroon et al.

2000; Ramula et al. 2008a; Burns et al. 2010). In our case, we

–0.4

–0.4

–0.2

–0.2

0.0

0.0

0.2

0.2

0.4

0.4

0.6

0.6

Stochastic population growth rate

Seq

uent

ial p

opul

atio

n gr

owth

rat

e

Common

Rare

–0.76

–0.27

–0.45

–0.67

–0.74

–0.3

Figure 7 Relationship between stochastic population growth rates

for randomly fluctuating (ks) and temporally autocorrelated

environments (sequential population growth rate, kseq) in relation

to species distribution (rare, common) for 26 species. The solid line

is an expected 1 : 1 relationship between the population growth

rates if ks and kseq were exactly equivalent. The six largest

deviations from the expected 1 : 1 are identified with their

autocorrelation co-efficient.



cannot exclude the possibility that variation in plant growth

is related to variation in population growth rate because

variance in survival and growth were correlated. As

expected, some of the variation in population growth rates

was also explained by fire and herbivory because both these

environmental factors can dramatically alter population

dynamics over a short period of time. Although the effect of

herbivory on plant population dynamics also depends on the

timing of herbivory and the sensitivity of population growth

rate to the life stages attacked (e.g. Ehrlen 2002; Maron &

Crone 2006; Ramula 2008), our findings suggest that both

fire and herbivory influenced plant population dynamics

largely through changes in plant survival as temporal

changes in fecundity explained little variation in population

growth rates.

Interestingly, population growth rates within populations

were negatively temporally autocorrelated, meaning that

good years were followed by bad years and vice versa.

Although both variation in survival and fecundity explained

negative autocorrelation in population growth rate, only

variation in survival contributed to temporal variation in

population growth rate. This indicates that fluctuating

survival was the main driver of variation and temporal

autocorrelation in plant population growth rates. Negative

temporal autocorrelation driven by survival may be related

to external (e.g. fire, herbivores) or internal (e.g. costs of

reproduction) factors that modify survival directly or

through correlations with the other vital rates.

Most stochastic population models assume that variation

is uncorrelated in time (Menges 2000; Pico & Riba 2002;

Kwit et al. 2004). However, a handful of studies point to the

potential importance of temporally autocorrelated stochas-

ticity (recently reviewed by Ruokolainen et al. 2009). These

studies have most often argued for positive autocorrelations

in population growth rates, due to resource storage, or

slower changes in the physical or biological environment that

buffer annual variation in environmental conditions (Halley

1996; Sabo & Post 2008). In contrast to both of these

expectations, we observed negatively autocorrelated popu-

lation growth rates. Negative temporal autocorrelation tends

to dampen environmental stochasticity, and slows down

extinction, resulting in a smaller extinction risk than positive

temporal autocorrelation or pure �white noise�. However, in

our dataset, realized population growth rates for ordered

environments were more often lower than expectations

without autocorrelations particularly for rare species with

large negative autocorrelations (Fig. 7). Therefore, the

temporal structure of variation in plant population dynamics

needs more attention in future research.

Although our dataset of 50 perennials is just a sample of

plant species and reflects any bias in the selection of species

for demographic studies, it includes species from temperate

ecosystems to the tropics and we therefore believe the

dataset allows generalizations to be made about plant

population dynamics. For instance, we can expect temporal

variation in survival rather than temporal variation in

fecundity to shape population dynamics of perennials, and

we can expect population growth rates to be temporally

autocorrelated. Departures from these generalizations may

occur for special cases, such as invasive species, although we

had insufficient invasive species to examine this in detail

here but see Ramula et al. (2008a). We encourage demog-

raphers to continue conducting demographic studies, and to

report detailed spatial and temporal information on their

study populations and environmental factors to enable

future comparative studies on plant population dynamics to

seek explanations for how populations respond to their

respective environments.

Consequences of realized temporal variation for plantpopulations

Deterministic population growth rates calculated from mean

matrices were generally good predictors of stochastic

population growth rates for randomly fluctuating environ-

ments, as others have also found (Boyce et al. 2006).

However, the inclusion of observed temporal autocorrela-

tion in population dynamics increased the difference

between deterministic and stochastic population growth

rates, and these differences were somewhat greater for rare

species and species with small to medium matrix dimen-

sions, where mean matrices usually produced more opti-

mistic estimates of population performance than the

stochastic methods. For our dataset, populations of rare

species with small sample sizes (e.g. 60–200 individuals for

Arabis fecunda, Astragalus scaphoides, Chamaecrista keyensis) were

often responsible for the greatest differences between

deterministic and stochastic population growth rates,

suggesting that the greater difference for rare species is

mainly because of larger sampling error. Larger sampling

error resulted from small sample sizes (c. < 300 individuals)

can lead to biased estimates of population growth rates

derived from matrix population models (Ramula et al. 2009).

Differences between deterministic and stochastic popula-

tion growth rates for autocorrelated environments are

important from a management point of view because small

differences in annual k estimates accumulate over time,

resulting in considerably different estimates of future

population size.

Lessons from variation in plant population dynamics

Our synthesis of population growth rates for perennial

plants has yielded important implications for understanding

and predicting plant population dynamics, showing that

environmental factors (e.g. fire and herbivory) strongly



influence population dynamics through space and time. The

lack of a phylogenetic signal on patterns of population

growth rates indicates that population dynamics can vary

greatly from species to species even within the same genus.

This makes it difficult to use demographic data from

close relatives (e.g. congeners) as a surrogate to produce

management recommendations when data for the target

species are lacking. The use of demographic data from

different populations within the same species would be

more appropriate, particularly if geographic distances among

populations are small and environmental factors similar.

However, simple space-for-time substitutions in plant

demographic analyses should be avoided. Greater temporal

variation than spatial variation in population growth rates

suggests that space-for-time data substitutions would

underestimate the true variation in population growth rates,

resulting in underestimated risk of extinction for declining

populations and consequently, over-optimistic predictions

of species performances.

Our results show that post-seedling survival is the key

vital rate contributing to temporal variation in population

growth rate and driving temporal autocorrelation of annual

population growth rates for perennial plants. Temporal

autocorrelation in population growth rates can rarely be

ignored in predictions of population dynamics as it occurs in

many plant populations and can change estimates of the

long-term population growth rate. For randomly fluctuating

environments realized variation in vital rates has a small

effect on plant population dynamics and mean matrix

and stochastic population growth rates are highly correlated,

although stochastic growth rates tend to be lower. However,

for temporally autocorrelated environments variation in

vital rates results in a slightly greater difference between

mean matrix and stochastic population growth rates. For

species with strong positive or negative autocorrelations,

stochastic population growth rates for temporally autocor-

related environments differed from those for randomly

fluctuating environments. This indicates that the sequence

of environmental conditions is essential for determining

plant population growth rate and should therefore be

incorporated into predictions of population dynamics.

Temporal autocorrelation in population dynamics necessi-

tates long-term observations over several years to produce

estimates of population performance, a finding that will be

particularly important in the face of future environmental

change.

A C K N O W L E D G E M E N T S

We thank the Australia–New Zealand Vegetation Function

Network for funding, Per Aronsson for database manage-

ment code, the editor, Mark Boyce and three anonymous

referees for helpful suggestions. YMB was funded by an

Australian Research Council Australian Research Fellowship

(DP0771387), SR by the Academy of Finland, JHB by

Tyson Research Center, the American Association of

University Women and the Center for Population Biology,

University of California, Davis.

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S U P P O R T I N G I N F O R M A T I O N

Additional Supporting Information may be found in the

online version of this article:

Figure S1 Phylogeny for 49 plant species used for

deterministic analyses.

Figure S2 Fitted population growth rates for all populations

of 49 plant species.

Table S1 Species included in the database.

Table S2 Model comparison with and without phylogeny for

log kdet.

Table S3 Parameter estimates for models of log kdet and

rlog kdet.

Table S4 Species with environmental factors.

Table S5 Herbivory levels for four species.

Table S6 Parameter estimates for the effect of temporal

autocorrelation on kseq.

As a service to our authors and readers, this journal provides

supporting information supplied by the authors. Such

materials are peer-reviewed and may be re-organized for

online delivery, but are not copy-edited or typeset. Technical

support issues arising from supporting information (other

than missing files) should be addressed to the authors.

Editor, James Grace

Manuscript received 7 April 2010

First decision made 10 May 2010

Manuscript accepted 20 May 2010



ecology letters, (2010) 13: 1182–1197 doi: 10.1111/j.1461 ... · mcmcglmm, population dynamics,...

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